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What is the slope of the line tangent to f (x) = X33x2 + 2x 4at the point (1, =4)?...

Question

What is the slope of the line tangent to f (x) = X33x2 + 2x 4at the point (1, =4)?

What is the slope of the line tangent to f (x) = X3 3x2 + 2x 4at the point (1, =4)?



Answers

Slope of a Tangent Line The equation of the tangent line of a function $f$ at (3,2) is $y=\frac{1}{3} x+1 .$ What is $f^{\prime}(3) ?$

With this tangent line problem. What we need to do is is no the point which they actually give us, the point is at one three halves. But then we also need the slope in the way that we find the slope is finding the directive and then we have to plug in the X coordinate. So the function that they give us is uh 3/2 X. Which actually confirms the point that we're looking at. But the first thing if we're doing the limit definition of the derivative, so we need to replace the X. And the problem with explosive H. Um it might help you, I don't know if this is actually helpful or not to distribute the two. And that problem to be to expose to H. But the next step would be to figure out what affects plus H minus F. Of X. Is. So we're doing the equation I just came up with, I'm going to use the simplified one because I don't know which one is better yet and in order to get the same denominator I'm actually just going to multiply them together and I'll worry about simplifying later and you can simplify just by multiplying the denominators together. I know there's a smaller common multiple there so you might do this a little bit faster if you factor out of two. But anyway uh so three times to actually give me six X. And I need to make sure that when I multiply the three by this, I also distribute it's really negative three times too. So it's negative six X. And negative three times to reach would be negative six H. And the whole point of doing that is to see that this will cancel. And when I go to take the limit as H approaches zero because this is the definition of the derivative. Um that answer I just came up with instead of dividing by H. I can actually multiply by the reciprocal and what you see then is these Hs eliminate and I can plug in zero right here and get a big zero over there. So I'm left with negative six over two X times to actually four X squared. That's what I meant earlier when I said that that could simplify, you know if you factored out it to earlier you would save a little bit of time there. Um But we still need to take this value for X and plug it in for the X that we had over here. Um And when you reduce that you get that the slope is negative three halves because one square still one. So why do we go through all that work is now we can use point slope form and you know why minus the white cordon equals the slope we just found and then X minus the exported, why corn exported. And some teachers will let you leave your equation on the tangent line like that. Other teachers might actually ask you to distribute. You have three have sex. This would be plus three halves, but then when I add another three halves to that, when I saw for why there would be six halves which reduces to three. Um so there is a slope intercept form. You might also have a teacher that doesn't like the fraction answers, so multiply everything by six and then start adding things to the left side and subtract six. You might have a teacher that prefers that answer as well, but all of them circled in green are correct.

Yeah. In the occasion given function is F x is equal to two X minus one. Multiplied by x square plus three. Now, in the kitchen it is asking to find the slope and tangent line. I had to a specified point. Even that is X is equal to one and Y is equal to zero. So for finding slope means I have to find at F gsx sorry, FDS X at X is equal to one and why is called to zero. Then after I will find the tangent line. So for finding FDS X I am using a rule that is called product rule. So according to this fool, the derivative of function means after sex will be equal to two X minus one, first turn right as it is then time self differentiation of with dy dx of X square plus three, then plus the sensation. Then try second term as it is means X squared plus three. Then take a differentiation of first time of gain function makes d by D X of two X minus one. So after so simplifying this, I will get FDA sex that is equal to two X minus one. No times of differentiation of X squared plus three will be two X. Then plus X squared plus three. And differentiation of two x minus one will be two. No we have to find FD sex at given point means put the point that is access call to one then floor philby if this one will be which is slow so value will be so for the value replaced the X by one. So after simplifying this I will get value duties. One then means two X minus one or two. In multiplied by one minus one will do one then two into one will you? Two plus X squared plus three means one plus three will be four times of year to So after solving this, I get final value that is after this one will be equal to 10. No, I am finding the tangent client. So as you know, the equation for tangent line will be, That is when the human point is excellent by one then the tenant line will be by minus Y. One is equal to slopes, means am multiplied by x minus X one. Support the value of Y one X one that is zero and X one will be one and put the value of em that will be 10 at given point. So my tenant line will be why minus zero is equal to 10, multiplied by x minus or the value of action that will be one. So that the engine line will be after simplifying this. I get value get equation dentist for tangent line, debt is why is you call to 10 x minus stan. So then then nine will be why is equal to 10 X minus stan. And the slope Ellie will be 10. So this is the answer. Thank you.

Okay, so we want to find the following equation to our attention sign. Given our function on our point, we know in order to find our spoke we need our slope at this point to find the equation over tangent line. So this year is like slope. So let's start by finding that so we have f prime of a late at C. C. Is our ex point of are given point, which is to So this is the limit as X approaches to of flex war that's three x squared, minus the function evaluated at two. So this is three times two squared. That's do we come for, which is 12 all over x minus to. So we see that in our numerator week. In fact, they're outs a three. So we're left with X squared minus four and this is a difference of two squares. That's X minus two times exports to small over X minus two. So let's cancel like tennis, and I want to use checks up. We have three times two plus two, which is equal to three times for what you 12. So this year is our slope. Okay, so we found our slope and we have our given point. So using points for the let's find the the equation to our tangent line. So this is everything times X minus X one. Okay, so we got why minus 12 is equal to 12 times x minus X one, which is still that's multiply through what we have. Okay, and now we have to do is add 12 to both sides to 12 to solve for white and our equation of or charging wine that's going to be equal to why is equal to 12 x minus 12.

Okay, so we want to find the equation over a tangent line. So we're going to need some point and our spoke and to find I slope, we can use this equation here. So let's start by finding art slope. So we have time at sea where C is our ex point. So that's one. So that's gonna be equal to the limit as X approaches one of our fallen function. Negative two X squared plus X minus three. And now we want to find F evaluated at one. So that's gonna be a native to plus a one minus three. So that gives us a negative four. So we have minus and that you don't force. That's plus for so we actually have a positive one here. So this is all over X minus one, and I Let's factor out our new Rita. So we convey activists into a negative two x, um, times something, and then an x. I'm something. We want this to be a positive Valium. So let's take negative one for both of these. So we get negative two x plus two x minus accident plus one. So that's good. And this is over. X minus one. And now we can cancel like times and I'll let's use ducks up. So we get negative two minus one, and that's equal to negative three. So the series are slow. So using point Slope form, let's plug in or slope and art points to find our equation over attention line. So we have why. Plus four is equal to negative three times X minus one. So that's equal to negative three X plus three. Now let's subtract for on both sides and we get that or equation of our tensions line that's going to be equal to y is equal to negative three x minus one


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